This is my exercise
Goals occur in a soccer game according to a Poisson Process. The average total number of goals scored in a 90-minute match is 2.68. Assume that two teams are evenly matched. Estimate the probability both teams will score the same number of goals.
And this is my attempt:
Assume that $X$ is the number of goals the first team score, and $Y$ is the number of goals the first team score. Since the two teams are evenly matched, then $X$ ~ Poiss(1.34) and $Y$ ~ Poiss(1.34) and they are independent. Then the probability that
\begin{align*} P(X=Y) = \sum\limits_{k=0}^\infty P(X=k).P(Y=k) = \sum\limits_{k=0}^\infty\dfrac{e^{-1.34}1.34^k}{k!}\dfrac{e^{-1.34}1.34^k}{k!} = e^{-2.68}\sum\limits_{k=0}^\infty\dfrac{1.34^{2k}}{(k!)^2} \end{align*}
Now I don't know how to compute the last series. Can somebody show me how to do so? Thank you.
